Efficiency and stability of sampling equilibrium in public goods games

• Author: Rajiv Sethi,Juan Camilo Cárdenas,César Mantilla
• DOI: http://doi.org/10.1111/jpet.12351
• Published date: 01 April 2020
• Date: 01 April 2020

Received: 30 September 2017

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Accepted: 24 November 2018

DOI: 10.1111/jpet.12351

ORIGINAL ARTICLE

Efficiency and stability of sampling equilibrium in

public goods games

César Mantilla

1

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Rajiv Sethi

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Juan Camilo Cárdenas

3

1

Department of Economics, Universidad del

Rosario, Bogotá, Colombia

2

Barnard College, Columbia University, New

York, NY

3

Department of Economics, Universidad de

los Andes, Bogotá, Colombia

Correspondence

César Mantilla, Department of Economics,

Universidad del Rosario, Bogotá 111711,

Colombia.

Email: cesar.mantilla@urosario.edu.co

Most models of social preferences and bounded ration-

ality that are effective in explaining efficiency‐increasing

departures from equilibrium behavior cannot easily

account for similar deviations when they are efficiency‐

reducing. We show that the notion of sampling equili-

brium, subject to a suitable stability refinement, can

account for behavior in both efficiency‐enhancing and

efficiency‐reducing conditions. In particular, in public

goods games with dominant strategy equilibria, stable

sampling equilibrium can involve the play of dominated

strategies with positive probability both when such

behavior increases aggregate payoffs (relative to the

standard prediction) and when it reduces aggregate

payoffs. The dominant strategy equilibrium prediction

changes abruptly from zero contribution to full contribu-

tion as a parameter crosses a threshold, whereas the

stable sampling equilibrium remains fully mixed through-

out. This is consistent with the available experimental

evidence.

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INTRODUCTION

Many important economic environments, including the private provision of public goods, common pool resource

extraction, team production, and intrahousehold resource allocation, often involve a tradeoff between efficiency

and equilibrium behavior. A significant experimental literature has explored the degree to which equilibrium

predictions are validated in laboratory settings, and models of bounded rationality and other‐regarding references

have been developed to account for observed departures from these predictions.

The experimental literature has generally avoided calibrations of these models in which there is no tradeoff

between equilibrium and efficiency, presumably because such cases have been considered trivial or uninteresting.

However, in the few instances in which this parameter space has been explored, the findings reveal very similar

divergences from equilibrium play. That is, players appear to deviate from equilibrium behavior in much the same

J Public Econ Theory. 2020;22:355–370. wileyonlinelibrary.com/journal/jpet © 2018 Wiley Periodicals, Inc.

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way regardless of whether such deviations raise or lower collective payoffs. Also, most models of social preferences

and bounded rationality that are effective in explaining efficiency‐increasing departures from equilibrium behavior

cannot easily account for similar deviations when they are efficiency reducing.

In this paper, we show that a particular model of procedural rationality based on the sampling of actions can

account for departures from standard equilibrium predictions in both the efficiency‐enhancing and efficiency‐

reducing cases.

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The model involves the independent sampling of each available action by players and the selection

of the action that yields the best realized payoff. These realized payoffs depend on random draws from a

distribution over the available actions. A sampling equilibrium is a distribution that is self‐replicating in the

following sense. The likelihood that a given action will be selected under the sampling procedure matches the

probability assigned to it in the incumbent distribution. This may be interpreted as the steady state of a model with

a large population of players with entry and exit in a manner described below.

We consider the predictions of this model for public goods games with dominant strategy equilibria that may or may

not be efficient and obtain the following results. When there are three or more players, the dominant strategy equilibrium

is unstable under the sampling dynamics regardless of whether or not it is efficient. As a result, strictly dominated

strategies are played with positive probability at any stable sampling equilibrium. The dominant strategy equilibrium

switches completely from zero contribution to full contribution as one moves from one regime to the other, whereas the

stable sampling equilibrium remains fully mixed throughout. This is consistent with the available experimental evidence.

In the two‐player case (with each player having at least three strategies) the results are somewhat different.

When the dominant strategy equilibrium is inefficient, it is also unstable under the sampling dynamics, so

dominated strategies are played with positive probability and efficiency is increased. But unlike the case of many

players, full contribution is stable under the sampling dynamics when it happens to also be efficient. Therefore, in

the two‐player case with an efficient dominant strategy equilibrium, the sampling approach generates the same

prediction as the standard approach.

The intuition underlying these findings may be roughly summarized as follows. Because realized payoffs are

determined by random draws from a population of actions, there is always a chance that a dominated strategy

yields a higher payoff than a dominant strategy when they are independently sampled. A dominant strategy

equilibrium will be unstable under the sampling dynamics if the deviation of a small share of the population away

from this strategy results in a sufficiently great likelihood that the dominant strategy will not yield the highest

realized payoff when sampled. Prior work has established that this instability can arise when the equilibrium is

inefficient; our results show that this can happen also when it is efficient.

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RELATED LITERATURE

Among economic environments in which voluntary contributions serve collectiveends are public goods (Bergstrom,

Blume, & Varian, 1986),common property (Ostrom, Walker, & Gardner,1992), team production (Holmstrom, 1982),

and allocation within families (Becker, 1981). The public goods case, in particular, has received a great deal of

attention from experimentalists, typically using specifications where contribution to the public good is efficient but

not individually rational (Andreoni, 1988; Chaudhuri, 2011; Isaac & Walker, 1988; Ledyard, 1995). A mismatch

between standardpredictions and experimentalfindings has given riseto several models of social preferences(Bolton

& Ockenfels, 2000; Fehr & Schmidt, 1999), reciprocity (Dufwenberg & Kirchsteiger, 2004; Fischbacher, Gächter, &

Fehr, 2001; Rabin, 1993), and bounded rationality (Andreoni, 1995; Aumann, 1997; Houser & Kurzban, 2002).

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The model of sampling equilibrium was developed by Osborne and Rubinstein (1998), and the stability refinement was introduced by Sethi (2000).

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Thereis also a substantialliteratureon mechanismsthat addressthe problemof underprovisionin publicgoods games; papersin this specialissue explore fixed

prizeraffles derivedfrom a biased contestsuccess function(Franke & Leininger,2019), commitmentsby playersto match contributionsby others (Buchholz&

Liu, 2019), and an endogenously chosen division of stakes between rounds in twice‐played cooperation dilemmas (Andreoni, K uhn, & Samuelson, 2019).

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MANTILLA ET AL.